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      Using the scale factor for enlarging an area

      Lesson details

      Learning outcome

      I can calculate the scale factor used to enlarge an area and use it to find missing lengths.

      Key learning points

      1. You can calculate the scale factor from the two areas.
      2. Having calculated the scale factor, you can find missing lengths.

      Keywords

      • Similar - Two shapes are similar if the only difference between them is their size. Their side lengths are in the same proportions.

      • Invariant - A property of a shape is invariant if that property has not changed after the shape is transformed.

      • Enlargement - Enlargement is a transformation that causes a change of size.

      • Scale factor - A scale factor is the multiplier between similar shapes that describes how large one shape is compared to the other.

      Common misconception

      A linear scale factor can only be found by finding the multiplicative relationship between two corresponding lengths.

      A linear scale factor can also be found by square rooting an area scale factor, where an area scale factor is found from the multiplicative relationship between the area of two similar 2D shapes, or corresponding faces on two similar 3D shapes.

      Teacher tip

      If a shape and another shape similar to it do not have right-angle markers on them, then you cannot assume that any angles are 90°, even if they look right-angled. Some examples in this lesson explicitly challenge pupils assuming that lengths can be found because angles in a shape look right-angled.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      A hexagon is enlarged by scale factor 3. What properties of the hexagon and its image are invariant?

      area
      Correct answer: interior angles
      Correct answer: order of rotational symmetry
      perimeter
      side lengths

      Q2.
      Rectangle A has been enlarged to create rectangle B. Select the correct statements.

      An image in a quiz
      The perimeter of rectangle A is $$\frac{1}{9} $$ the perimeter of rectangle B
      Correct answer: The perimeter of rectangle A is $$\frac{1}{3}$$ the perimeter of rectangle B
      The area of rectangle A is $$\frac{1}{3}$$t he area of rectangle B
      Correct answer: The area of rectangle A is $$\frac{1}{9}$$ the area of rectangle B
      The area of rectangle A is $$3$$ the area of rectangle B

      Q3.
      Rectangle A has been enlarged to create rectangle B. The area of rectangle B is cm²

      An image in a quiz
      Correct Answer: 405

      Q4.
      Trapezium B is an enlargement of trapezium A. Calculate the area of trapezium B.

      An image in a quiz
      40 cm²
      120 cm²
      240 cm²
      Correct answer: 360 cm²
      400 cm²

      Q5.
      Match each linear scale factor to the area scale factor.

      Correct Answer:Linear scale factor: $$5$$,Area scale factor: $$25$$
      Correct Answer:Linear scale factor: $$\frac{1}{2}$$,Area scale factor: $$\frac{1}{4}$$
      Correct Answer:Linear scale factor: $$3$$,Area scale factor: $$9$$
      Correct Answer:Linear scale factor: $$0.1$$,Area scale factor: $$0.01$$
      Correct Answer:Linear scale factor: $$10$$,Area scale factor: $$100$$
      Correct Answer:Linear scale factor: $$\frac{3}{2}$$,Area scale factor: $$\frac{9}{4}$$

      Q6.
      Shape S is an enlargement an enlargement of shape R. The area of shape R is 24 cm². The area of shape S is cm².

      An image in a quiz
      Correct Answer: 150

      6 Questions

      Q1.
      The transformation that causes a change in size of the object is .

      Correct Answer: enlargement

      Q2.
      Rectangles A and B are similar. What is the area scale factor from A to B?

      An image in a quiz
      $$\frac{1}{16}$$
      $$\frac{1}{4}$$
      4
      Correct answer: 16
      64

      Q3.
      Shapes P and Q are similar. The perimeter of shape P is 20 cm and the perimeter of shape Q is 40 cm. The area scale factor from P to Q is .

      An image in a quiz
      Correct Answer: 4

      Q4.
      Sectors E and F are similar. The area of sector E is 25π cm². What is the area of sector F?

      An image in a quiz
      31π cm²
      50π cm²
      75π cm²
      Correct answer: 100π cm²
      125π cm²

      Q5.
      Sectors E and F are similar. The area of shape E is 26π cm² and the area of sector F is 58.5π cm². The radius of sector F is cm.

      An image in a quiz
      Correct Answer: 9, 9 cm

      Q6.
      Triangles L and M are similar. The area of triangle M is 150 cm². What is the perimeter of triangle M?

      An image in a quiz
      36 cm
      48 cm
      Correct answer: 60 cm
      72 cm
      75 cm

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